Is there a computer vision service for classifying images on a fixed array of images provided by me?
Actually I'm trying the same thing with the Azure Computer Vision API. Although the API is very good in identifying objects, it has problems identifying specific consumer products (in my experience though). For example it can't really distinguish between two pair of shoes, or two pair of watches. 

People recommended me using the: Custom Vision Service combined with Custom Decision Service. I haven't really looked in to them both. But maybe it's useful for you.